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gibson:teaching:fall-2016:math753:norms-orthogonality

Math 753/853 Norms, inner products, and orthogonality

Ok, this is a big set of topics, and nothing I've found covers the topic at the right level of detail or depth. So, here's a summary of a few key points you should understand. These were spelled out in detail during lecture.

Inner product

The inner product of two vectors and

where is the transpose of . If , and are orthogonal.

2-norm

The 2-norm of a vector is defined as

Note that .

The 2-norm of a matrix is defined as

You can think of as the maximum amplification factor in length that can occur under the map .

Orthogonal matrices

A matrix is an orthogonal matrix if its inverse is its transpose: . The columns of an orthogonal matrix are a set of orthogonal vectors.

Key properties of orthogonal matrices:

• The inner product is preserved under orthogonal transformations: .
• The vector 2-norm is preserved under orthogonal transformations: .
• The matrix 2-norm is preserved under orthogonal transformations: .
• The 2-norm of an orthogonal matrix is one: .